Nonlocal Differential Operators Research Articles

Fredholm properties of nonlocal differential operators via spectral flow

We establish Fredholm properties for a class of nonlocal differential operators. Using mild convergence and localization conditions on the nonlocal terms, we also show how to compute Fredholm indices via a generalized spectral flow, using crossing numbers of generalized spatial eigenvalues. We illustrate possible applications of the results in a nonlinear and a linear setting. We first prove the existence of small viscous shock waves in nonlocal conservation laws with small spatially localized source terms. We also show how our results can be used to study edge bifurcations in eigenvalue problems using Lyapunov-Schmidt reduction instead of a Gap Lemma.

Adaptive Moving Mesh Finite Element Method for Space Fractional Advection Dispersion Equation

Space fractional advection dispersion equation includes non-local differential operators, which leads to calculating numerical integrals with weakly singular kernel on every subintervals. Oscillations are visible in computed solutions when fractional order tends to 1. Studies show that the stiffness matrix of time semi-discretization can be calculated directly by formulas established from a special variational formulation. Oscillations are eliminated by using adaptive moving mesh and De Boor algorithm, while the number of nodes remains unchanged.

Edges and diffractive effects in Casimir energies

The prototypical Casimir effect arises when a scalar field is confined between parallel Dirichlet boundaries. We study corrections to this when the boundaries themselves have apertures and edges. We consider several geometries: a single plate with a slit in it, perpendicular plates separated by a gap, and two parallel plates, one of which has a long slit of large width, related to the case of one plate being semi-infinite. We develop a general formalism for studying such problems, based on the wavefunctional for the field in the gap between the plates. This formalism leads to a lower dimensional theory defined on the open regions of the plates or boundaries. The Casimir energy is then given in terms of the determinant of the nonlocal differential operator which defines the lower dimensional theory. We develop perturbative methods for computing these determinants. Our results are in good agreement with known results based on Monte Carlo simulations. The method is well suited to isolating the diffractive contributions to the Casimir energy.

Existence and Exponential Decay of Solutions to a Quasilinear Thermoelastic Plate System

We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in Rn, n ≤ 3. Existence of finite energy solutions describing the dynamics of a nonlinear thermoelastic plate is established. In addition asymptotic long time behavior of weak solutions is discussed. It is shown that finite energy solutions decay exponentially to zero with the rate depending only on the (finite energy) size of initial conditions. The proofs are based on methods of weak compactness along with nonlocal partial differential operator multipliers which supply the sought after “recovery” inequalities. Regularity of solutions is also discussed by exploiting the underlying analyticity of the linearized semigroup along with a related maximal parabolic regularity [1, 16, 44].

Further spectral properties of uniformly elliptic operators that include a non-local term

In this paper we consider the eigenvalues of a class of non-local differential operators. The real eigenvalues of certain examples of these operators have previously been plotted (as functions of a fundamental parameter in the operator), using appropriate computer software. Here we show how the complex eigenvalues of such operators may also be plotted. We use the plots of both the real and the complex eigenvalues to motivate a number of new theoretical results regarding the structure of the spectrum, which we subsequently prove under suitable hypotheses.

Spectral properties of non-local differential operators

In this article we consider the spectral properties of a class of non-local operators that arise from the study of non-local reaction-diffusion equations. Such equations are used to model a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded perturbations of linear (local) differential operators. The non-local perturbation is in the form of an integral term. It is shown here that the spectral properties of these non-local operators can differ considerably from those of their local counterpart. Multiplicities of eigenvalues are studied and new oscillation results for the associated eigenfunctions are presented. These results highlight problems with certain similar results and provide an alternative formulation. Finally, the stability of steady states of associated non-local reaction-diffusion equations is discussed.

Staggered mesh for the anisotropic and viscoelastic wave equation

Computation of the spatial derivatives with nonlocal differential operators, such as the Fourier pseudospectral method, may cause strong numerical artifacts in the form of noncausal ringing. This situation happens when regular grids are used. The problem is attacked by using a staggered pseudospectral technique, with a different scheme for each rheological relation. The nature and causes of acausal ringing in regular grid methods and the reasons why staggered‐grid methods eliminate this problem are explained in papers by Fornberg (1990) and Özdenvar and McMechan (1996). Thus, the objective here is not to propose a new method but to develop the algorithm for the viscoelastic and transversely isotropic (VTI) wave equation, for which the technique can be implemented without interpolation. The algorithm is illustrated for one physical situation that requires very high accuracy, such as a fluid‐solid interface, where very large contrasts in material properties occur. The staggered‐grid solution is noise free in the dynamic range where regular grids generate artifacts that may have amplitudes similar to those of physical arrivals.

Lattice space-time from Poincaré and κ-Poincaré algebras

Introducing a dimensional parameter κ we propose a realization of the Poincaré and κ-Poincaré algebras by non-local space-time differential operators. The Klein-Gordon and Dirac equations take the form of a finite difference equation. The connection with the deformed Heisenberg algebra is discussed leading to a generalized uncertainty relation.

Lattice QCD as a theory of interacting surfaces

Pure gauge lattice QCD at arbitrary D is considered. Exact integration over link variables in an arbitrary D-volume leads naturally to an appearance of a set of surfaces filling the volume and gives an exact expression for functional of their boundaries. The interaction between each two surfaces is proportional to their common area and is realized by a non-local matrix differential operator acting on their boundaries. The surface self-interaction is give by the QCD 2 functional of boundary. Partition functions and observables (Wilson loop averages) are written as an averages over all configurations of an integer-valued field living on a surfaces.

On the Gelfand–Dickey algebra GD(SLn) and the W n-symmetry. I. The bosonic case

The algebra Ξ of nonlinear (local and nonlocal) differential operators, acting on the ring of analytic functions, is studied. It is shown in particular that this space splits into 3×2 special subalgebras ∑jr, j=0,±1, r=±1. Each subalgebra is completely specified by quantum numbers s and (p,q) describing the conformal spin, and the lowest and the highest degrees, respectively. The algebra ∑++ (and its dual ∑−−) of local (pure nonlocal) differential operators is used to calculate the general expression of the Gelfand–Dickey bracket and the Wn-symmetry Poisson, one in terms of a set of spin j canonical fields uj, 2≤j≤n and a nonlinear u-cubic dependent differential operator D(n,i,j;z,u). The explicit form of this operator is worked out. Other remarkable features are also discussed.

Motion picture of nerve impulse propagation using computer animation.

ArticleMotion picture of nerve impulse propagation using computer animation.R FitzhughR FitzhughPublished Online:01 Nov 1968https://doi.org/10.1152/jappl.1968.25.5.628MoreSectionsPDF (579 KB)Download PDF ToolsExport citationAdd to favoritesGet permissionsTrack citations ShareShare onFacebookTwitterLinkedInWeChat Previous Back to Top Next Download PDF FiguresReferencesRelatedInformation Cited ByMathematical and Circuit Level Analysis Interpretation and Recommendations on Neuron Models7 May 2022 | Journal of Circuits, Systems and Computers, Vol. 31, No. 12Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems I: Well-Posedness14 July 2021 | Journal of Dynamics and Differential Equations, Vol. 34, No. 2Fitzhugh–Nagumo Model12 June 2022Exponential dichotomies for nonlocal differential operators with infinite range interactionsJournal of Differential Equations, Vol. 301Travelling wave solutions for fully discrete FitzHugh-Nagumo type equations with infinite-range interactionsJournal of Mathematical Analysis and Applications, Vol. 502, No. 2Analogy circuit synthesis and dynamics confirmation of a bipolar pulse current-forced 2D Wilson neuron model5 June 2021 | The European Physical Journal Special Topics, Vol. 230, No. 7-8Chemomechanical origin of directed locomotion driven by internal chemical signalsScience Advances, Vol. 6, No. 18Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations11 February 2020Traveling Waves for Spatially Discrete Systems of FitzHugh--Nagumo Type with Periodic Coefficients21 August 2019 | SIAM Journal on Mathematical Analysis, Vol. 51, No. 4ReferencesSpatiotemporal Patterns of Noise-Driven Confined Actin Waves in Living Cells27 January 2017 | Physical Review Letters, Vol. 118, No. 4Retrograde and Direct Wave Locomotion in a Photosensitive Self-Oscillating Gel13 October 2016 | Angewandte Chemie, Vol. 128, No. 46Retrograde and Direct Wave Locomotion in a Photosensitive Self-Oscillating Gel13 October 2016 | Angewandte Chemie International Edition, Vol. 55, No. 46The effect of process delay on dynamical behaviors in a self-feedback nonlinear oscillatorCommunications in Nonlinear Science and Numerical Simulation, Vol. 39The derivation of continuum limits of neuronal networks with gap-junction couplingsNetworks and Heterogeneous Media, Vol. 9, No. 1FitzHugh–Nagumo Model1 April 2014Complex order van der Pol oscillator12 November 2010 | Nonlinear Dynamics, Vol. 65, No. 3Accurate computation of the stable solitary wave for the FitzHugh-Nagumo equationsJournal of Mathematical Biology, Vol. 13, No. 3Pulse evolution on coupled nerve fibresBulletin of Mathematical Biology, Vol. 43, No. 4Mathematical models of cardiac arrhythmias (spiral waves)Pharmacology & Therapeutics. Part B: General and Systematic Pharmacology, Vol. 3, No. 4 More from this issue > Volume 25Issue 5November 1968Pages 628-30 https://doi.org/10.1152/jappl.1968.25.5.628PubMed5687371History Published online 1 November 1968 Published in print 1 November 1968 Metrics